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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.04867 |
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| _version_ | 1866915774457708544 |
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| author | Gomes, Paulo Henrique Cunha |
| author_facet | Gomes, Paulo Henrique Cunha |
| contents | We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots,60\}$ such that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ is a covering (dominating set) of the Johnson graph $J(60,6)$ with covering radius $3$ in the Johnson metric. The construction is purely combinatorial, based on a fixed split of $[60]$ into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to $[2m]$ (with $m$ even). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04867 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A 910-block explicit construction guaranteeing a triple intersection with every $6$-subset of $[60]$ Gomes, Paulo Henrique Cunha Combinatorics We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots,60\}$ such that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ is a covering (dominating set) of the Johnson graph $J(60,6)$ with covering radius $3$ in the Johnson metric. The construction is purely combinatorial, based on a fixed split of $[60]$ into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to $[2m]$ (with $m$ even). |
| title | A 910-block explicit construction guaranteeing a triple intersection with every $6$-subset of $[60]$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.04867 |